Reduction method for solving Boundary Value Problem on Graph to it's internal edge

Abstract

The main aim to this research is to reduce the boundary value problem for fourth differential equation on geometric graph with cycles to a problem on a internal edge provided that the right hand side of the differential equation is identically zero on some subgraph of the original graph, and in this research we find the sign of some coefficients in the boundary conditions of the reduced problem and relationship between these coefficients. That's helping us to prove existence and uniqueness for a boundary value problem resulting from this reduction.

In order to reach our desired goal, we study the reduction method of boundary value problem for fourth differential equation on tree geometric graph(no cycles), finally we can say that our research help us to study green function on edge(interval) instead of complex sty ding on geometric graph(with cycles).